How to solve system of equations containing summation over variable to solve for?

Question

How do we solve for $\pi_i$ in the following? $$\pi_i=\frac{\sum\limits_j N_{i,j}}{\sum\limits_j\left( \ell_{i,j} \frac{\sum\limits_k N_{k,j}}{\sum\limits_k \ell_{k,j} \pi_k}\right)}\qquad\forall i,\qquad \sum\limits_i \pi_i=1.$$

Answer 1

Assume $\pi_i$ where $i \in \{1,2,\cdots ,I\}$, you have $I$ equations in the form you gave plus the the equation with sum over $\pi_i$, so you can remove on of the $I$ equations, then you will have: $$ A *\Pi = b $$ where $A$ is the $I\times I$ identity matrix with the last row where all elements are replaced by ones, $\Pi$ is the vector you are solving for with size $I\times 1$, and $b$ is a vector with size $I\times 1$ with the elements on the right except the last element is $1$.

Then, you can solve it by any technique you prefer.